"We have books."
Translation:Nous avons des livres.
French uses articles a bit differently than in English. With some exceptions, French nouns usually have to have an article before them, even in cases where no article is used in English. So in this case, because we have multiple books that are not specified, we have to use the indefinite plural article "des". You can think of this as meaning "some" in English; "We have some books" would have the same meaning as "We have books".
In other contexts, French omits articles where English uses articles. For example, "I am a doctor" is "Je suis médecin", not "Je suis un médecin".
No - the present tense conjugations of "avoir" are:
There are no exceptions to these conjugations, so whenever the subject is "nous", you can only use "avons". Similarly, if the subject is "ils" or "elles", you can only use "ont".
This answer has a brief explanation (and also the difference in pronounciation)